Lecture over Section 3-1 of Barnett's "Finite Mathematics."

1. Math 1300 Finite Mathematics
Section 3.1 Simple Interest
Jason Aubrey
Department of Mathematics
University of Missouri
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Jason Aubrey Math 1300 Finite Mathematics

2. Definition (Simple Interest)
I = Prt
where
I = interest
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Jason Aubrey Math 1300 Finite Mathematics

3. Definition (Simple Interest)
I = Prt
where
I = interest
P = principal
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Jason Aubrey Math 1300 Finite Mathematics

4. Definition (Simple Interest)
I = Prt
where
I = interest
P = principal
r = annual simple interest rate (written as a decimal)
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Jason Aubrey Math 1300 Finite Mathematics

5. Definition (Simple Interest)
I = Prt
where
I = interest
P = principal
r = annual simple interest rate (written as a decimal)
t = time in years
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Jason Aubrey Math 1300 Finite Mathematics

6. Example: A department store charges 21% for overdue
accounts. How much interest will be owed on a $650 account
that is 3 months overdue?
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Jason Aubrey Math 1300 Finite Mathematics

7. Example: A department store charges 21% for overdue
accounts. How much interest will be owed on a $650 account
that is 3 months overdue?
Here we are given that P = $650, r = 0.21, and t = 3/12.
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Jason Aubrey Math 1300 Finite Mathematics

8. Example: A department store charges 21% for overdue
accounts. How much interest will be owed on a $650 account
that is 3 months overdue?
Here we are given that P = $650, r = 0.21, and t = 3/12.
I = Prt
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Jason Aubrey Math 1300 Finite Mathematics

9. Example: A department store charges 21% for overdue
accounts. How much interest will be owed on a $650 account
that is 3 months overdue?
Here we are given that P = $650, r = 0.21, and t = 3/12.
I = Prt
3
I = 650(0.21) = 34.13
12
Therefore $34.13 is owed.
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Jason Aubrey Math 1300 Finite Mathematics

10. Example: A commercial for a loan company states, “You only
pay $0.16 a day for each $600 borrowed.” If you borrow $1800
for 240 days, what amount will you repay, and what annual
interest rate is the company actually charging?
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Jason Aubrey Math 1300 Finite Mathematics

11. Example: A commercial for a loan company states, “You only
pay $0.16 a day for each $600 borrowed.” If you borrow $1800
for 240 days, what amount will you repay, and what annual
interest rate is the company actually charging?
You pay a fee of 3 × $0.16 = $0.48 per day.
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Jason Aubrey Math 1300 Finite Mathematics

12. Example: A commercial for a loan company states, “You only
pay $0.16 a day for each $600 borrowed.” If you borrow $1800
for 240 days, what amount will you repay, and what annual
interest rate is the company actually charging?
You pay a fee of 3 × $0.16 = $0.48 per day.
This is a total fee of $0.48 × 240 = $115.20 over the life of
the loan.
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Jason Aubrey Math 1300 Finite Mathematics

13. Example: A commercial for a loan company states, “You only
pay $0.16 a day for each $600 borrowed.” If you borrow $1800
for 240 days, what amount will you repay, and what annual
interest rate is the company actually charging?
You pay a fee of 3 × $0.16 = $0.48 per day.
This is a total fee of $0.48 × 240 = $115.20 over the life of
the loan.
So, I = $115.20, P = $1800, and t = 240/360.
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Jason Aubrey Math 1300 Finite Mathematics

14. Now we apply our simple interest formula...
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Jason Aubrey Math 1300 Finite Mathematics

15. Now we apply our simple interest formula...
I = Prt
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Jason Aubrey Math 1300 Finite Mathematics

16. Now we apply our simple interest formula...
I = Prt
240
$115.20 = ($1, 800)r
360
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Jason Aubrey Math 1300 Finite Mathematics

17. Now we apply our simple interest formula...
I = Prt
240
$115.20 = ($1, 800)r
360
r = 0.096
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Jason Aubrey Math 1300 Finite Mathematics

18. Now we apply our simple interest formula...
I = Prt
240
$115.20 = ($1, 800)r
360
r = 0.096
So, you repay a total of $1,915.20 and the annual interest rate
is 9.6%.
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Jason Aubrey Math 1300 Finite Mathematics

19. Theorem (Amount: Simple Interest)
A = P + Prt
= P(1 + rt)
A = amount, or future value
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Jason Aubrey Math 1300 Finite Mathematics

20. Theorem (Amount: Simple Interest)
A = P + Prt
= P(1 + rt)
A = amount, or future value
P = principal, or present value
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Jason Aubrey Math 1300 Finite Mathematics

21. Theorem (Amount: Simple Interest)
A = P + Prt
= P(1 + rt)
A = amount, or future value
P = principal, or present value
r = annual simple interest rate (written as a decimal)
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Jason Aubrey Math 1300 Finite Mathematics

22. Theorem (Amount: Simple Interest)
A = P + Prt
= P(1 + rt)
A = amount, or future value
P = principal, or present value
r = annual simple interest rate (written as a decimal)
t = time in years
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Jason Aubrey Math 1300 Finite Mathematics

23. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
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Jason Aubrey Math 1300 Finite Mathematics

24. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here
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Jason Aubrey Math 1300 Finite Mathematics

25. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37,
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Jason Aubrey Math 1300 Finite Mathematics

26. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000,
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Jason Aubrey Math 1300 Finite Mathematics

27. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000, and t = 13 52
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Jason Aubrey Math 1300 Finite Mathematics

28. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000, and t = 13 52
A = P(1 + rt)
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Jason Aubrey Math 1300 Finite Mathematics

29. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000, and t = 13 52
A = P(1 + rt)
13
$1, 000 = $989.37 1 + r
52
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Jason Aubrey Math 1300 Finite Mathematics

30. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000, and t = 13 52
A = P(1 + rt)
13
$1, 000 = $989.37 1 + r
52
13
1.011 ≈ 1 + r
52
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Jason Aubrey Math 1300 Finite Mathematics

31. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000, and t = 13 52
A = P(1 + rt)
13
$1, 000 = $989.37 1 + r
52
13
1.011 ≈ 1 + r
52
13
0.011 ≈ r
52
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Jason Aubrey Math 1300 Finite Mathematics

32. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000, and t = 13 52
A = P(1 + rt)
13
$1, 000 = $989.37 1 + r
52
13
1.011 ≈ 1 + r
52
13
0.011 ≈ r
52
r ≈ 0.044
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Jason Aubrey Math 1300 Finite Mathematics

33. Example: What annual interest rate is earned by a 13-week
T-bill with a maturity value of $1,000 that sells for $989.37?
In T-bill problems P corresponds to the selling price ( or
purchase price) of the T-bill, and A corresponds to the maturity
value. Here P = $989.37, A = $1, 000, and t = 13 52
A = P(1 + rt)
13
$1, 000 = $989.37 1 + r
52
13
1.011 ≈ 1 + r
52
13
0.011 ≈ r
52
r ≈ 0.044 or 4.4%
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Jason Aubrey Math 1300 Finite Mathematics

34. Example: Many investment firms charge commissions on
transactions based on the amount of the transaction. Suppose
that an investment firm charges commissions on stock trades
according to the following commission schedule:
Transaction Size Commission
Under $3,000 $25+1.8% of principal
$3000-$10,000 $37 + 1.4% of principal
Over $10,000 $107 + 0.7% of principal
Suppose an investor purchases 175 shares at $15.00 a share,
holds the stock for 26 weeks, and then sells the stock for
$17.25 per share. Find the annual interest rate earned by this
investment.
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35. Step 1: Find the total cost of the purchase.
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Jason Aubrey Math 1300 Finite Mathematics

36. Step 1: Find the total cost of the purchase.
$15.00(175) = $2,625 - Principal
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Jason Aubrey Math 1300 Finite Mathematics

37. Step 1: Find the total cost of the purchase.
$15.00(175) = $2,625 - Principal
$25 + $2,625(0.018) = $72.25 - Commission
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Jason Aubrey Math 1300 Finite Mathematics

38. Step 1: Find the total cost of the purchase.
$15.00(175) = $2,625 - Principal
$25 + $2,625(0.018) = $72.25 - Commission
$2,625 + $72.25 = $2,697.25 - Total cost of purchase
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Jason Aubrey Math 1300 Finite Mathematics

39. Step 2: Find the net revenue from the sale.
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Jason Aubrey Math 1300 Finite Mathematics

40. Step 2: Find the net revenue from the sale.
$17.25(175) = $3,018.75 - Principal
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Jason Aubrey Math 1300 Finite Mathematics

41. Step 2: Find the net revenue from the sale.
$17.25(175) = $3,018.75 - Principal
$37 + ($3,018.75)(0.014) = $79.26 -Commission
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Jason Aubrey Math 1300 Finite Mathematics

42. Step 2: Find the net revenue from the sale.
$17.25(175) = $3,018.75 - Principal
$37 + ($3,018.75)(0.014) = $79.26 -Commission
$3,018.75 - $79.26 = $2,939.49 - Net revenue from sale
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Jason Aubrey Math 1300 Finite Mathematics

43. Step 3: Calculate annual interest rate:
,
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44. Step 3: Calculate annual interest rate:
Here A = $2, 939.49, ,
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Jason Aubrey Math 1300 Finite Mathematics

45. Step 3: Calculate annual interest rate:
Here A = $2, 939.49, P = $2697.25,
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Jason Aubrey Math 1300 Finite Mathematics

46. Step 3: Calculate annual interest rate:
Here A = $2, 939.49, P = $2697.25, and t = 26/52
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Jason Aubrey Math 1300 Finite Mathematics

47. Step 3: Calculate annual interest rate:
Here A = $2, 939.49, P = $2697.25, and t = 26/52
A = P(1 + rt)
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48. Step 3: Calculate annual interest rate:
Here A = $2, 939.49, P = $2697.25, and t = 26/52
A = P(1 + rt)
2939.49 = 2697.25(1 + r (.5))
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Jason Aubrey Math 1300 Finite Mathematics

49. Step 3: Calculate annual interest rate:
Here A = $2, 939.49, P = $2697.25, and t = 26/52
A = P(1 + rt)
2939.49 = 2697.25(1 + r (.5))
r = 0.1796
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Jason Aubrey Math 1300 Finite Mathematics

50. Example: Suppose that after buying a new car you decide to
sell your old car to a friend. You accept a 270-day note for
$3,500 at 10% simple interest as payment. (Both principal and
interest will be paid at the end of 270 days.) Sixty days later
you find that you need the money and sell the note to a third
party for $3,550. What annual interest rate will the third party
recieve for the investment? (Express the answer as a
percentage, correct to three decimal places).
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51. Step 1: Find the amount that will be paid at the end of 270
days to the holder of the note:
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Jason Aubrey Math 1300 Finite Mathematics

52. Step 1: Find the amount that will be paid at the end of 270
days to the holder of the note:
A = P(1 + rt)
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Jason Aubrey Math 1300 Finite Mathematics

53. Step 1: Find the amount that will be paid at the end of 270
days to the holder of the note:
A = P(1 + rt)
270
= ($3, 500) 1 + (0.1)
360
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Jason Aubrey Math 1300 Finite Mathematics

54. Step 1: Find the amount that will be paid at the end of 270
days to the holder of the note:
A = P(1 + rt)
270
= ($3, 500) 1 + (0.1)
360
= $3, 762.50
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55. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
.
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Jason Aubrey Math 1300 Finite Mathematics

56. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
A = $3, 762.50,
.
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Jason Aubrey Math 1300 Finite Mathematics

57. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
A = $3, 762.50, P = $3, 550,
.
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Jason Aubrey Math 1300 Finite Mathematics

58. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
210
A = $3, 762.50, P = $3, 550, t = 360
.
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Jason Aubrey Math 1300 Finite Mathematics

59. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
210
A = $3, 762.50, P = $3, 550, t = 360
.
A = P(1 + rt)
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Jason Aubrey Math 1300 Finite Mathematics

60. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
210
A = $3, 762.50, P = $3, 550, t = 360
.
A = P(1 + rt)
210
$3, 762.50 = $3, 550 1 + r
360
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Jason Aubrey Math 1300 Finite Mathematics

61. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
210
A = $3, 762.50, P = $3, 550, t = 360
.
A = P(1 + rt)
210
$3, 762.50 = $3, 550 1 + r
360
r = 0.10262
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Jason Aubrey Math 1300 Finite Mathematics

62. Step 2: For the third party, we are to find the annual rate of
interest r required to make $3,550 grow to $3,762.50 in 210
days (270 - 60). So we need to find r given that:
210
A = $3, 762.50, P = $3, 550, t = 360
.
A = P(1 + rt)
210
$3, 762.50 = $3, 550 1 + r
360
r = 0.10262 or 10.262%.
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Jason Aubrey Math 1300 Finite Mathematics

63. Example: Many tax preparation firms offer their clients a refund
anticipation loan (RAL). For a fee, the firm will give the client his
refund when the return is filed. The loan is repaid when the IRS
sends the refund directly to the firm. Thus, the RAL fee is
equivalent to the interest charge for the loan. The schedule
below is from a major RAL lender.
RAL Amount RAL Fee
0-$500 $29.00
$501-$1,000 $39.00
$1,001-$1,500 $49.00
$1,501-$2,000 $69.00
$2,001-$2,500 $89.00
A client recieves a $480 RAL which is repaid in 25 days. What
is the annual interest rate for this loan?
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64. First, P = $480.
.
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Jason Aubrey Math 1300 Finite Mathematics

65. First, P = $480. So according to the schedule, the amount
charged is I = $29.00. .
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Jason Aubrey Math 1300 Finite Mathematics

66. First, P = $480. So according to the schedule, the amount
25
charged is I = $29.00. t = 360 .
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Jason Aubrey Math 1300 Finite Mathematics

67. First, P = $480. So according to the schedule, the amount
25
charged is I = $29.00. t = 360 . We now compute r :
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Jason Aubrey Math 1300 Finite Mathematics

68. First, P = $480. So according to the schedule, the amount
25
charged is I = $29.00. t = 360 . We now compute r :
I = Prt
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Jason Aubrey Math 1300 Finite Mathematics

69. First, P = $480. So according to the schedule, the amount
25
charged is I = $29.00. t = 360 . We now compute r :
I = Prt
25
29 = (480)r
360
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Jason Aubrey Math 1300 Finite Mathematics

70. First, P = $480. So according to the schedule, the amount
25
charged is I = $29.00. t = 360 . We now compute r :
I = Prt
25
29 = (480)r
360
r = 0.87
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Jason Aubrey Math 1300 Finite Mathematics